The Mathematics of Smoothed Particle Hydrodynamics (SPH) Consistency

Since its inception Smoothed Particle Hydrodynamics (SPH) has been widely employed as a numerical tool in different areas of science, engineering, and more recently in the animation of fluids for computer graphics applications. Although SPH is still in the process of experiencing continual theoretical and technical developments, the method has been improved over the years to overcome some shortcomings and deficiencies. Its widespread success is due to its simplicity, ease of implementation, and robustness in modeling complex systems. However, despite recent progress in consolidating its theoretical foundations, a long-standing key aspect of SPH is related to the loss of particle consistency, which affects its accuracy and convergence properties. In this paper, an overview of the mathematical aspects of the SPH consistency is presented with a focus on the most recent developments.